ISI Interview Question

Algebra Level 4

M = ∑ c y c x , y , z x a y + b z M=\large{\sum^{x, \ y, \ z}_{cyc} \dfrac{x}{ay+bz}}

Find the minimum value of M M , if a + b = 7 a+b=7 and a , b , x , y , z ∈ R + a,b,x,y,z \in \mathbb{R}^{+}


The answer is 0.428.

Md Zuhair by Md Zuhair , 20, India

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1 solution

Given expression is :

M = x a y + b z + y a z + b x + z a x + b y = x 2 a x y + b x z + y 2 a y z + b x y + z 2 a z x + b y z M = \dfrac{x}{ay+bz} + \dfrac{y}{az+bx} + \dfrac{z}{ax+by} = \dfrac{x^{2}}{axy+bxz} + \dfrac{y^{2}}{ayz+bxy} + \dfrac{z^{2}}{azx+byz}

Using Titu's Lemma :

M ≥ [ x + y + z ] 2 ( a + b ) ( y x + z y + x z ) M \ge \dfrac{[x+y+z]^{2}}{(a+b)(yx+zy+xz)}

M ≥ x 2 + y 2 + z 2 7 ( x y + y z + z x ) + 2 7 M \ge \dfrac{x^{2} + y^{2} +z^{2}}{7(xy+yz+zx)} + \dfrac{2}{7}

Now, using the result x 2 + y 2 + z 2 ≥ x y + y z + z x x^{2} + y^{2} + z^{2} \ge xy + yz + zx which follows by rearrangement inequality we get :

M ≥ 3 7 M \ge \dfrac{3}{7}

5 Helpful 1 Interesting 1 Brilliant 0 Confused

I know, When i learnt Titu's Lemma first, I was obsessed with it :P

Everyone likes to do questions with it :)

Md Zuhair - 3 years, 8 months ago

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pls give the source of these interview questions of isi @Md Zuhair .

rajdeep brahma - 3 years ago

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Ok. Why is your WhatsApp off? Kemon rank Holo JEE ADV E? Ar this source is from a senior

Md Zuhair - 3 years ago

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